Apart from the analytic theories, computational simulations and numerical methods have offered effective ways to probe supercoiling dynamics. Atomistic molecular dynamics (MD) simulations offer the finest details of such dynamics. For example, MD simulations of DNA mini-circles (\(10^{2}\)  \(b\) ) reveal writhe fluctuation and configuration diversity[1]}, [2]}, [3]}. To improve computational efficiency, a coarse-grained model that treats nucleotides as beads with three interaction sites, called oxDNA, was developed[4]}. Furthermore, by unitizing MD and the coarse-grained simulations, the multi-scale dynamics of supercoiling have been studied[5]}, [6]}. Nevertheless, computational costs of these models are high and the simulation time scales are limited by microseconds[7]}. The models without considering DNA sequence structure are capable of exceeding such a limit. An elasto-dynamic model (i.e., the Kircholff rod), for example, has been developed, by which DNA is characterized in terms of a continuum rod while omitting thermal fluctuations[8]}. The corresponding applications were carried out on supercoil removal[9]}, and on compressed DNA inside bacteriophage cavity to allow DNA ejection[10]}. In particular, for a representative polymer physics model, the numerical approach of the WLC model, i.e., the discrete worm-like chain (dWLC) method, has been widely implemented in simulating DNA with the thermal fluctuations incorporated[11]}. For example, the Monte Carlo (MC) simulations of the WLC model have been used in describing DNA thermodynamics, conformation and site juxtaposition[12]}, [13]}, [14]}. The Brownian dynamics (BD) simulations, on the other hand, have been applied more widely in studying supercoiling[11]}, [16]}. Such type of studies successfully show the plectoneme diffusion[17]}, supercoiling conformations [12]}, [19]}, buckling transition[20]}, [21]}, [22]}, modeling a DNA wrapped around a model histone[23]}, and supercoil removal by nicking[24]}. Besides, other semi-flexible polymer models with Lennard-Jones potential have also been utilized to investigate the mechanism for supercoiling by rotating the ends[25]}, or using interwoven and braided polymers to produce plectonemes[26]}.
